Semide nite programming relaxations for the graph partitioning problem (

A new semide nite programming, SDP, relaxation for the general graph partitioning problem, GP, is derived. The relaxation arises from the dual of the (homogenized) Lagrangian dual of an appropriate quadratic representation of GP. The quadratic representation includes a representation of the 0,1 constraints in GP. The special structure of the relaxation is exploited in order to project onto the minimal face of the cone of positive-semide nite matrices which contains the feasible set. This guarantees that the Slater constraint quali cation holds, which allows for a numerically stable primal–dual interior-point solution technique. A gangster operator is the key to providing an e cient representation of the constraints in the relaxation. An incomplete preconditioned conjugate gradient method is used for solving the large linear systems which arise when nding the Newton direction. Only dual feasibility is enforced, which results in the desired lower bounds, but avoids the expensive primal feasibility calculations. Numerical results illustrate the e cacy of the SDP relaxations. ? 1999 Elsevier Science B.V. All rights reserved. MSC: 90C25; 90C27; 15A48